CYCLES OF LENGTH-0 MODULO-4 IN GRAPHS

In several papers a variety of questions have been raised concerning t he existence of cycles of length 0 mod k in graphs. For the case k = 4 , we answer three of these questions by showing that a graph G contain s such a cycle provided it has any of the following three properties: (1) G has minimum degree at least 2 and at most two vertices of degree 2, (2) G is not 3-colorable, and (3) G is a subdivision of a graph of order p greater-than-or-equal-to 5 with at least 3p - 5 edges.